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A084386
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Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n-1) + 3*a(n-3), with a(0) = a(1) = a(2) = 1.
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10
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1, 1, 1, 4, 7, 10, 22, 43, 73, 139, 268, 487, 904, 1708, 3169, 5881, 11005, 20512, 38155, 71170, 132706, 247171, 460681, 858799, 1600312, 2982355, 5558752, 10359688, 19306753, 35983009, 67062073, 124982332, 232931359, 434117578, 809064574
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OFFSET
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0,4
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COMMENTS
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This comment covers an infinite family of growth sequences, where a(n) = a(n-1) + k*a(n-m). k is number of pairs per litter and m is periods until adulthood. G.f. = 1/(1-x-k*x^m). For example, A000930 has k=1 and m=3 while A006130 has k=3 and m=2.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 3, 4*a(n-3) equals the number of 4-colored compositions of n with all parts >= 3, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+2) equals the number of words of length n on alphabet {0,1,2,3}, having at least two zeros between every two successive nonzero letters. - Milan Janjic, Feb 07 2015
Number of compositions of n into one sort of part 1 and three sorts of part 3 (see the g.f.). - Joerg Arndt, Feb 07 2015
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet , J. Int. Seq. 19 (2016) # 16.1.3, Example 9
Merrill Jensen, Generating Functions
Index entries for linear recurrences with constant coefficients, signature (1,0,3).
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FORMULA
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a(n) = a(n-1) + 3*a(n-3). a(n) = A052900(n+3)/3.
G.f.: 1/(1-x-3*x^3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-2*k, k)*3^k. - Paul Barry, Nov 18 2003
G.f.: W(0)/2, where W(k) = 1 + 1/(1 - x*(1 + 3*x^2)/(x*(1 + 3*x^2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
Starting (1 + x + 4*x^2 + ...), is the INVERT transform of (1 + 3*x^2). - Gary W. Adamson, Mar 27 2017
a(m+n) = a(m)*a(n) + 3*a(m-1)*a(n-2) + 3*a(m-2)*a(n-1). - Michael Tulskikh, Jun 23 2020
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MAPLE
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seq(add(binomial(n-2*k, k)*3^k, k=0..floor(n/3)), n=0..34); # Zerinvary Lajos, Apr 03 2007
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MATHEMATICA
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a[0]=a[1]=a[2]=1; a[n_] := a[n]=a[n-1]+3a[n-3]; Table[a[n], {n, 0, 34}]
LinearRecurrence[{1, 0, 3}, {1, 1, 1}, 37] (* Robert G. Wilson v, Jul 12 2014 *)
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PROG
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(PARI) a(n)=([0, 1, 0; 0, 0, 1; 3, 0, 1]^n*[1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
(MAGMA) I:=[1, 1, 1]; [n le 3 select I[n] else Self(n-1)+3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 28 2017
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CROSSREFS
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Partial sums of A052900. Also A052900/3.
Cf. A000930, A001045, A006130.
Sequence in context: A227686 A161863 A102649 * A275176 A024726 A024948
Adjacent sequences: A084383 A084384 A084385 * A084387 A084388 A084389
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KEYWORD
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easy,nonn
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AUTHOR
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Merrill Jensen (mpjensen(AT)mninter.net), Jun 23 2003
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EXTENSIONS
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Edited by Dean Hickerson, Jun 24 2003
Recurrence appended to the name by Antti Karttunen, Mar 28 2017
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STATUS
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approved
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